Definition and position
Big Idea: The median is the middle value when data is ordered.
Half the values are below it, half above.
Not affected by outliers.
Key difference: Median ≠ Mean when data is skewed.
Median is resistant to outliers; mean is not.
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Median from raw data
Odd count: use middle: Data: 3, 5, 7, 9, 11 (n=5) Position = (5+1)/2 = 3rd value Median = 7
Even count: average two middle: Data: 3, 5, 7, 9 (n=4) Position = (4+1)/2 = 2.5 → between 2nd and 3rd Median = (5 + 7)/2 = 6
Worked example — median of test scores
Ten students scored: 65, 72, 58, 80, 75, 68, 90, 71, 77, 84.
Find the median.
Step by step
- Order the data from smallest to largest.
- n = 10, so position = (10+1)/2 = 5.5 — between the 5th and 6th values.
- 5th value = 72; 6th value = 75. Median = average of the two.
Final answer
Median = 73.5.
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Median from cumulative frequency
Use CF graph or table: Find position n/2. Read from cumulative frequency graph (ogive) or table at that position.
Example: n=100, position=50 Read cumulative frequency table to find value where CF=50.
Worked example — median from cumulative frequency
A cumulative frequency table for 50 students' test scores is shown below.
| Marks | 20 | 30 | 40 | 50 |
|---|---|---|---|---|
| Cumulative frequency | 8 | 22 | 38 | 50 |
Estimate the median.
Step by step
- Find the median position.
- Locate where cumulative frequency reaches 25. It is between 22 (at 30 marks) and 38 (at 40 marks), so the median is in the class [30, 40).
- Use linear interpolation inside the median class. The class [30,40) needs to add 25 − 22 = 3 more values, out of 38 − 22 = 16 in this class.
- Check: 31.9 lies inside [30, 40).
Final answer
Median ≈ 31.9 marks.
When to use median
Use median when: Data contains outliers Distribution is skewed You want the most typical value Working with ranked/ordinal data
Compare mean vs median: If mean >> median → data skewed right (outliers high) If median >> mean → data skewed left (outliers low) If mean ≈ median → data roughly symmetric